March 28, 2006
Multi-rank Capon Beamforming
Beamforming is widely used in wireless communications, radar, sonar, seismology, and radio astronomy to localize sources in space. In standard Capon beamforming, the assumption is that the signal of interest has a rank-one covariance matrix, with a known structure, corresponding to an ideal propagation (not necessarily plane) from a point source to a receiving array.
There are many cases where a known rank-one covariance does not model reality. Rather, a reasonable model is that consecutive array snapshots produce signals that are drawn from a multi-dimensional subspace and the corresponding signal covariances are either multi-rank, or /rank-one but unknown. Such scenarios arise frequently in wireless communications, and radar and sonar imaging, due to multi-path, random scattering, random changes in the propagation medium, and flexing arrays. In such cases, mismatches between the actual signal model and the one assumed by the standard Capon beamformer degrade the performance.
In this talk, we first present a systematic treatment of different wavefront models, and their corresponding subspace and covariance models. We then derive two multi-rank generalizations of the Capon beamformer: (1) a matched subspace beamformer, for the case where the signal of interest is drawn from a multi-dimensional subspace and has a multi-rank covariance matrix, to account for non-coherent multi-path, local and random scattering, and random changes in the propagation medium, and (2) a matched direction beamformer, for the case where the signal of interest is drawn from an unknown one-dimensional subspace and has an unknown rank-one covariance matrix, to account for coherent multi-path and array calibration errors. As we show, multi-rank Capon beamformers may be obtained in a unified way by solving a linearly constrained quadratic minimization problem, and then designing a constraint matrix that amounts for a resolution onto a dominant subspace (for matched direction beamforming) or a subdominant subspace (for matched subspace beamforming). Finally, we present numerical examples to demonstrate that multi-rank Capon beamformers can detect and resolve wavefronts of interest.
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